Book•
Classical recursion theory
01 Jan 1989-
TL;DR: Theories of Recursive functions, Hierarchies of recursive functions, and Arithmetical sets: Recursively enumerable sets.
Abstract: Preface. Introduction. Theories of Recursive functions. Hierarchies of recursive functions. Recursively enumerable sets. Recursively enumerable degrees. Limit sets. Arithmetical sets. Arithmetical degrees. Enumeration degrees. Bibliography. Notation index. Subject index.
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Book•
01 Jan 2006TL;DR: In this paper, the authors provide a comprehensive treatment of the problem of predicting individual sequences using expert advice, a general framework within which many related problems can be cast and discussed, such as repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems.
Abstract: This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.
3,615 citations
Book•
12 Mar 2014TL;DR: This book provides a solid fundament for studying various aspects of computability and complexity in analysis and is written in a style suitable for graduate-level and senior students in computer science and mathematics.
Abstract: Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.
1,330 citations
Book•
01 Jan 2009
TL;DR: This book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.
Abstract: The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. Covering the basics as well as recent research results, this book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.
638 citations
Cites background from "Classical recursion theory"
...Solutions to most of the exercises in this Chapter can be found in Soare (1987) or Odifreddi (1989). We only include solutions to selected exercises....
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...For proofs, see for instance Odifreddi (1989). A structure (U,≤,∨) is an uppersemilattice if (U,≤) is a partial order and, for each x, y ∈ U , x ∨ y is the least upper bound of x, y 1....
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...Things left out here can be found in Soare (1987) or Odifreddi (1989). After covering the basics, we will return to the issues outlined above....
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Proceedings Article•
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26 Jun 2000
TL;DR: This work determines the complexity of model checking and query evaluation on automatic structures for fragments of first-order logic and gives model-theoretic characterisations for automatic structures via interpretations.
Abstract: We study definability and complexity issues for automatic and /spl omega/-automatic structures. These are, in general, infinite structures but they can be finitely presented by a collection of automata. Moreover they admit effective (in fact automatic) evaluation of all first-order queries. Therefore, automatic structures provide an interesting framework for extending many algorithmic and logical methods from finite structures to infinite ones. We explain the notion of (/spl omega/-)automatic structures, give examples, and discuss the relationship to automatic groups. We determine the complexity of model checking and query evaluation on automatic structures for fragments of first-order logic. Further we study closure properties and definability issues on automatic structures and present a technique for proving that a structure is not automatic. We give model-theoretic characterisations for automatic structures via interpretations. Finally we discuss the composition theory of automatic structures and prove that they are closed under finitary Feferman-Vaught-like products.
365 citations
TL;DR: In this system self-maintaining organizations arise as a generic consequence of two features of chemistry, without appeal to natural selection, and are held as calling for increased attention to the structural basis of biological order.
335 citations
Cites background from "Classical recursion theory"
...…models (e.g., Turing machines rest on the notion of a state transition, A-calculus rests on the notion of substitution, Post (or Thue) systems are aimed at recursive enumerability, distributed state transitions are central to cellular automata, and so forth; for an overview see Odifreddi, 1989)....
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